It is also worthy to note that in general, slow manifolds described by Theorem REF  are non-unique. This fact, which is also true for Fenichel slow manifolds in the ODE setting and for center-type manifolds more generally [1]}, [2]}, [3]}, follows from the use of \(C^\infty \)  cutoff functions in order to control the dynamics entering or leaving a neighbourhood of \(S\)  along the center directions. As a consequence, any two slow (resp. Fenichel, center) manifolds must have the same Taylor series in \(\varepsilon \) , however they may still be separated by \(O(e^{-c/\varepsilon })\)  distances beyond all orders, where \(c>0\) . Because of the closeness of slow manifolds, we shall frequently refer to any fixed choice of slow manifold \(S_{\varepsilon }\)  described by Theorem REF  as `the' slow manifold, as has become common nomenclature in fast-slow ODE theory.
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